Largest 'hole' in a Sudoku; Largest 'emtpy space'

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Largest 'hole' in a Sudoku; Largest 'emtpy space'

Postby tso » Sun May 28, 2006 5:34 pm

On a similar vein of this thread:

'Loop' of clues with 45 cell 'hole'.
Code: Select all
+-------+-------+-------+
| . 6 3 | 5 7 . | 4 2 . |
| 5 . . | . . 3 | . . 7 |
| 1 . . | . . . | . . 9 |
+-------+-------+-------+
| . 5 . | . . . | . . 4 |
| 8 . . | . . . | . . 1 |
| 6 . . | . . . | . 3 . |
+-------+-------+-------+
| 3 . . | . . . | . . 5 |
| 2 . . | 9 . . | . . 6 |
| . 7 9 | . 6 5 | 8 1 . |
+-------+-------+-------+

All clues form a single connected chain, each adjacent to exactly two others, either diagonally or orthogonally.

This example has 90 degree rotational symmetry and the hole is therefore centered.

When creating a valid puzzle:

What is the largest hole that can be produced while maintaining at least some symmetry?
What is the largest hole that can be produced ignoring symmetry?
Can the size of the hole in a valid puzzle be increased by allowing clues to be adjacent to more than 2 cells?



Sudoku with 61 cell 'emtpy space'
Code: Select all
+-------+-------+-------+
| . . 9 | 1 . . | . . . |
| . 8 7 | 2 . . | . . . |
| . . . | . . . | . 5 2 |
+-------+-------+-------+
| . . . | . . . | 1 2 9 |
| . . . | . . . | . . . |
| 4 5 6 | . . . | . . . |
+-------+-------+-------+
| 9 3 . | . . . | . . . |
| . . . | . . 5 | 3 4 . |
| . . . | . . 4 | 7 . . |
+-------+-------+-------+

An 'empty space' is an area of any shape that has no clues in it and does not surround any clues or group of clues completely.

This example has 180 degree rotational symmetry.

What is the largest empty space that can be made, with/without some symmetry?
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Postby Chessmaster » Sun May 28, 2006 5:42 pm

wow once i saw a puzzle that had two squares that where totaly empty
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Postby tarek » Sun May 28, 2006 5:47 pm

Chessmaster wrote:wow once i saw a puzzle that had two squares that where totaly empty


There are puzzles with 3 & 4 empty boxes........

The Superior thread has some examples....

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Postby tso » Sun May 28, 2006 6:24 pm

Chessmaster wrote:wow once i saw a puzzle that had two squares that where totaly empty


See this post for puzzles with 4 empty boxes, 5 empty 3x3 areas, as well as puzzles with 9 empty groups.
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Re: Largest 'hole' in a Sudoku; Largest 'emtpy space'

Postby JPF » Sun May 28, 2006 9:14 pm

tso wrote:'Loop' of clues with 45 cell 'hole'.

an other pattern, fully symmetric :
Code: Select all
 . 1 5 | 8 . 9 | 6 7 .
 2 . . | . 4 . | . . 8
 8 . . | . . . | . . 1
-------+-------+-------
 5 . . | . . . | . . 6
 . 8 . | . . . | . 9 .
 4 . . | . . . | . . 5
-------+-------+-------
 9 . . | . . . | . . 2
 1 . . | . 5 . | . . 3
 . 4 2 | 3 . 1 | 8 5 .



here is a '46 holes' with symmetry :

Code: Select all
 . 7 1 | 3 4 5 | 6 8 .
 4 . . | . . . | . . 7
 6 . . | . . . | . . 3
-------+-------+-------
 9 . . | . . . | . . 6
 . 2 . | . . . | . 7 .
 7 . . | . . . | . . 4
-------+-------+-------
 1 . . | . . . | . . 5
 3 . . | . 6 . | . . 8
 . 4 6 | 7 . 8 | 3 1 .


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Re: Largest 'hole' in a Sudoku; Largest 'emtpy space'

Postby JPF » Mon May 29, 2006 11:07 pm

tso wrote:Sudoku with 'emtpy space'
An 'empty space' is an area of any shape that has no clues in it and does not surround any clues or group of clues completely.
...
What is the largest empty space that can be made, with/without some symmetry?


Here's a 64 cells 'empty space' picked from the gfroyle list :

Code: Select all
 1 . . | . 5 . | 2 . .
 3 7 . | . . . | . . .
 . 6 . | . . 9 | . . .
-------+-------+-------
 . . 8 | . 2 . | 4 . .
 . . . | 3 . . | . 6 .
 . . . | . . . | . 1 .
-------+-------+-------
 . . 5 | . . . | 9 . .
 . . . | 7 . . | . . .
 . . . | 6 . . | . . .


Is it what we are looking for ?

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Re: Largest 'hole' in a Sudoku; Largest 'emtpy space'

Postby tso » Mon May 29, 2006 11:28 pm

[deleted on a count of temporary blindness]
Last edited by tso on Mon May 29, 2006 8:28 pm, edited 1 time in total.
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Postby JPF » Mon May 29, 2006 11:55 pm

does it look better like this ?

Code: Select all
 1 . . . 5 . 2 . .
 3 7 . . . . . . .
 . 6 . . . 9 . . .
 . . 8 . 2 . 4 . .
 . . . 3 . . . 6 .
 . . . . . . . 1 .
 . . 5 . . . 9 . .
 . . . 7 . . . . .
 . . . 6 . . . . .


JPF
[Edit : minor typo corrected]
Last edited by JPF on Tue May 30, 2006 2:01 am, edited 1 time in total.
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Postby tso » Tue May 30, 2006 12:29 am

JPF wrote:does it look better like this ?


Why, yes, it does -- and that's obviously the maximum without symmetry. A 63 cell space with symmetry will probably not be rare.

[EDIT]

Here's one with 61:

Code: Select all
 . 6 9 . . . . . .
 . . . 9 . . . . .
 . . . 1 8 . . 2 .
 . . . . . . 3 . 6
 . . 7 . . . 5 . .
 8 . 1 . . . . . .
 . 5 . . 3 6 . . .
 . . . . . 5 . . .
 . . . . . . 7 1 .
Last edited by tso on Tue May 30, 2006 2:56 pm, edited 1 time in total.
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Postby JPF » Tue May 30, 2006 7:56 am

a 64 cell space with 8 groups of clues (instead of 4) :

Code: Select all

 2 . 6 . . . 1 . .
 . . . 3 . . . 7 .
 . . . 4 . . . . .
 3 . . 5 . . . . 2
 . 4 . . . . 6 . .
 . . . . . . 7 . .
 7 . . . . . . 3 .
 . . . . 6 . . . 8
 . . . . 1 . . . .


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Postby JPF » Tue May 30, 2006 6:01 pm

Now, the next question could be :
Find the simple closed loop of clues which has the minimum empty space in it .

Here's a symmetric puzzle with 17 cell 'hole' in it :

Code: Select all

 . . . | . . . | . . .
 . 3 . | . . . | . 4 .
 8 . 9 | . . . | 5 . 3
-------+-------+-------
 5 . 7 | . . . | 6 . 8
 6 . 2 | . . . | 3 . 1
 1 . . | 2 . 8 | . . 7
-------+-------+-------
 . 7 . | . 9 . | . 1 .
 . . 5 | . . . | 8 . .
 . . . | 1 3 4 | . . .



or to avoid any dazzling effect ...

Code: Select all
 . . . . . . . . .
 . 3 . . . . . 4 .
 8 . 9 . . . 5 . 3
 5 . 7 . . . 6 . 8
 6 . 2 . . . 3 . 1
 1 . . 2 . 8 . . 7
 . 7 . . 9 . . 1 .
 . . 5 . . . 8 . .
 . . . 1 3 4 . . .


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Postby tso » Tue May 30, 2006 7:18 pm

Good one. Here's one with 16:

Code: Select all
. . . 8 6 7 . . .
. . 3 . . . 5 . .
. . . 5 . 1 . . .
. . . 1 . 2 . . .
. . 9 . . . 3 . .
. 6 . . 9 . . 8 .
. 8 . 3 . 5 . 4 .
. 7 . 4 . 8 . 1 .
. . 5 . . . 7 . .


... and an asymmetrical one with 14:

Code: Select all
. . . . 8 7 . . .
. . . 6 . . 4 . .
. . . 5 . 1 . . .
. . . 1 . 3 . . .
. . 8 . . . 9 . .
. 1 . . 2 . . 5 .
. 6 . 7 . 2 . 1 .
. . 4 . . 8 . 9 .
. . . . . . 6 . .
Last edited by tso on Tue May 30, 2006 3:44 pm, edited 1 time in total.
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Postby tso » Tue May 30, 2006 7:38 pm

... and if cells in the simple loop are allowed to touch three others:


12 cell hole with vertical mirror symmetry:
Code: Select all
. . 2 5 7 4 8 . .
. . 5 . . . 9 . .
. . 6 3 . 9 4 . .
. . . 7 . 2 . . .
. . . 1 . 3 . . .
. 2 4 6 . 5 1 7 .
. 9 . . . . . 4 .
. 7 8 9 6 1 2 3 .
. . . . . . . . .


12 cell hole -- loop has 180 degree symmetry, puzzle has none:
Code: Select all
. . 4 9 7 2 6 1 .
. . 3 . . . . 8 .
. . 9 8 . 3 4 2 .
. . . 5 . 4 . . .
. . . 1 . 7 . . .
. 9 8 2 . 6 7 . .
. 2 . . . . 5 . .
. 6 7 4 2 5 3 . .
. . . . . . . . .


[EDIT]

Oh, this is much better:


9 cell hole, 180 degree symmetry:

Code: Select all
. . . . . . . . .
. . . . 9 4 1 2 .
. . . 5 7 . . 4 .
. . . 8 . . 2 9 .
. . 5 9 . 3 6 . .
. 3 8 . . 7 . . .
. 9 . . 8 5 . . .
. 2 7 1 6 . . . .
. . . . . . . . .
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Postby JPF » Tue May 30, 2006 8:18 pm

tso wrote:... and if cells in the simple loop are allowed to touch three others:

I think we should keep the first definition

[deleted ; pattern not valid]
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Postby JPF » Tue May 30, 2006 10:43 pm

This one is 15, symmetric and romantic...

Code: Select all

 . . 1 . . . 3 . .
 . 5 . 7 . 4 . 6 .
 . 4 . 8 . 6 . 7 .
 . 6 . . 4 . . 8 .
 . . 8 . 2 . 1 . .
 . . 3 . . . 2 . .
 . . . 1 . 5 . . .
 . . . 2 . 8 . . .
 . . . . 7 . . . .


but, I'm not sure it's a valid one.

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